Improved WKB Phase Shift Via Virtual Turning Points for Waveguides with Index Discontinuities,

The WKB method has been widely used to obtain approximate solutions of the guided mode propagation constants for optical waveguides with slowly varying graded-index profiles. One of the key features of the standard method, which is properly applied only to problems with isolated turning points, is the assumption of a constant phase shift Pi4 at a turning Point. However, it has long been known that when turning points are not isolated, but are instead near some other transition points, the phase shift is no longer constant. Here, we consider the presence of an index discontinuity near or at a turning point in an optical waveguide and present analytical expressions of the improved phase shift. The principle of this phase shift correction is based upon two main points. The first consists of using Airy functions as the field solutions at the discontinuity.